Statistical analysis of cointegration vectors

The estimation and testing of long-run relations in economic modeling are addressed. Starting with a vector autoregressive (VAR) model, the hypothesis of cointegration is formulated as the hypothesis of reduced rank of the long-run impact matrix. This is given in a simple parametric form that allows the application of the method of maximum likelihood and likelihood ratio tests. In this way, one can derive estimates and test statistics for the hypothesis of a given number of cointegration vectors, as well as estimates and tests for linear hypotheses about the cointegration vectors and their weights. The asymptotic inferences concerning the number of cointegrating vectors involve nonstandard distributions. Inference concerning linear restrictions on the cointegration vectors and their weights can be performed using the usual chi squared methods. In the case of linear restrictions on beta, a Wald test procedure is suggested. The proposed methods are illustrated by money demand data from the Danish and Finnish economies.

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Maximum likelihood estimation and inference on cointegration — with applications to the demand for money

Maximun likelihood estimation and inference on cointegration - With applications to the demand for money

Presents the likelihood methods for the analysis of cointegration in VAR models with Gaussian errors, seasonal dummies, and constant terms. Discusses likelihood ratio tests of cointegration rank and find the asymptotic distribution of the test statistics. Shows that the asymptotic distribution of the maximum likelihood estimator is mixed Gausssian.

Estimation and hypothesis testing of cointegration vectors in Gaussian vector autoregressive models / Søren Johansen

Using research designs patterned after randomized experiments, many recent economic studies examine outcome measures for treatment groups and comparison groups that are not randomly assigned. By using variation in explanatory variables generated by changes in state laws, government draft mechanisms, or other means, these studies obtain variation that is readily examined and is plausibly exogenous. This paper describes the advantages of these studies and suggests how they can be improved. It also provides aids in judging the validity of inferences they draw. Design complications such as multiple treatment and comparison groups and multiple pre- or post-intervention observations are advocated.

Comparing Predictive Accuracy

This study is devoted to the analysis of multivariate time series data. Such data might arise in business and economics, engineering, geophysical sciences, agriculture, and many other fields. The emphasis is on providing an account of the basic concepts and methods which are useful in analyzing such data. The book presupposes a familiarity with univariate time series as might be gained from one term of a graduate course, but it is otherwise self-contained. It covers the basic topics such as autocovariance matrices of stationary processes, vector ARMA models and their properties, forecasting ARMA processes, least squares and maximum likelihood estimation techniques for vector AR and ARMA models, and associated likelihood ratio testing procedures for model building. In addition, it presents more advanced topics and techniques including reduced rank structure, structural indices, scalar component models, canonical correlation analyses for vector time series, multivariate nonstationary unit root models and co-integration structure, and state-space models and Kalman flltering techniques.

Elements of Multivariate Time Series Analysis

Introduction to Mathematical Statistics (4th ed.).

. The nonstationary multivariate autoregressive (AR) model Φ (L)Yt=et is considered for an m-dimensional process {Yt}, where it is assumed that det {Φ(L)}= 0 has d< m unit roots and all other roots are outside the unit circle, and also that rank {Φ(1)}=r (r=m–d). Limiting distribution results obtained by Ahn and Reinsel for the least-squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite-sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.

Vector autoregressive models with unit roots and reduced rank structure:estimation. likelihood ratio test, and forecasting

For an m-variate (“partially”) nonstationary vector autoregressive process {Y t }, we consider the autoregressive model Φ(L)Y t = e t , where Φ(L) = I − Φ1 L − … − Φ p L p and det{Φ(z)} = 0 has d < m roots equal to unity and all other roots are outside the unit circle. It is also assumed that rank {Φ(1)} = r, r = m − d, so that each component of the first differences W t = Y t − Y t − 1 is stationary. The relation of the model to error correction models and co-integration (Engle and Granger 1987) is discussed. The process {Y t } has the error correction representation Φ*(L)(1 − L)Y t = −P 2(I r − Λ r )Q 2 1 Y t-1 + e t , where Q(I m − Φ(1))P = J = diag(I d , Λ r ) is in Jordan canonical form and Q' = [Q 1, Q 2]. It follows that the transformation Z t = QY t = [Zt 1t , Z2 2t ] t is such that the d × 1 process Z 1t is nonstationary with Z 1t − Z 1t-1 stationary while Z 2t is stationary. Asymptotic distribution theory for least squares parameter estimators of the model is first considered. A Gaussia...

Gregory C. Reinsel

Papers

Elements of Multivariate Time Series Analysis

Introduction to Mathematical Statistics (4th ed.).

Vector autoregressive models with unit roots and reduced rank structure:estimation. likelihood ratio test, and forecasting

Estimation for Partially Nonstationary Multivariate Autoregressive Models

Prediction of multivariate time series by autoregressive model fitting